Analysis of the Primal-Dual Central Path for Nonlinear Semidefinite Optimization Without the Nondegeneracy Condition
研究了非线性半定优化问题中心路径的存在性与唯一性，在无退化条件下证明了路径存在且对偶分量收敛到拉格朗日乘子集的解析中心，并明确了牛顿方程唯一可解的区域。
We study properties of the central path underlying a nonlinear semidefinite optimization problem, called an NSDP for short. The latest radical work on this topic was contributed by Yamashita and Yabe [Yamashita H, Yabe H (2012) Local and superlinear convergence of a primal-dual interior point method for nonlinear semidefinite programming. Mathematical Programming 132(1-2):1–30]: they proved that the Jacobian of a certain equation system derived from the Karush–Kuhn–Tucker (KKT) conditions of the NSDP is nonsingular at a KKT point under the second-order sufficient condition (SOSC), the strict complementarity condition (SC), and the nondegeneracy condition (NC). This yields uniqueness and existence of the central path through the implicit function theorem. In this paper, we consider the following three assumptions on a KKT point: the strong SOSC, the SC, and the Mangasarian–Fromovitz constraint qualification. Under the absence of the NC, the Lagrange multiplier set is not necessarily a singleton, and the nonsingularity of the above-mentioned Jacobian is no longer valid. Nonetheless, we establish that the central path exists uniquely, and moreover prove that the dual component of the path converges to the so-called analytic center of the Lagrange multiplier set. As another notable result, we clarify a region around the central path where Newton’s equations relevant to primal-dual interior-point methods are uniquely solvable. Funding: This work was supported by the Japan Society for the Promotion of Science [Grant-in-Aid for Young Scientists 20K19748 and Grant-in-Aid for Scientific Research (C) 20H04145]. The Japan Society for the Promotion of Science [Grant-in-Aid for Scientific Research (C) 25K15008].